Tabular methods and function approximation are two approaches to solving reinforcement learning (RL) problems, differing primarily in how they represent and update value estimates. Tabular methods store exact value estimates (like Q-values or state-values) for every possible state-action pair in a lookup table, making them precise but impractical for large or continuous state spaces. Function approximation replaces the table with a parameterized function (e.g., a neural network) that generalizes across states, sacrificing exactness for scalability. The key distinction is that tabular methods work explicitly with discrete, enumerable states, while function approximation handles complexity by learning patterns from data.
Tabular methods excel in small, discrete environments where all states can be explicitly tracked. For example, in a grid world with 10x10 states, a Q-table with 100 entries (one per state) can store the expected reward for each action. Algorithms like Q-Learning or SARSA update these values directly: when the agent visits a state, it adjusts the corresponding table entry based on observed rewards and future estimates. However, this approach becomes infeasible in real-world problems. A robot with sensor data (e.g., continuous joint angles or camera pixels) has infinitely many states, making a table impossible to store or update. Even moderately complex environments, like board games with (10^{50}) states, exceed computational limits. Tabular methods also struggle with partial observability—if the agent can’t distinguish states precisely, the table becomes unreliable.
Function approximation addresses scalability by replacing the table with a model that predicts values. For instance, a neural network can take a state (e.g., game screen pixels) as input and output Q-values for each action. Deep Q-Networks (DQN) use this approach, training the network to minimize prediction errors via gradient descent. Linear regression, decision trees, or tile coding are simpler alternatives. The trade-off is approximation error: the model might misestimate values for under-sampled states. However, it generalizes better—learning that "states with similar features have similar values"—enabling handling of high-dimensional inputs like images or sensor streams. For example, in AlphaGo, function approximation (via CNNs) generalized patterns from board positions to unseen states. Challenges include balancing exploration-exploitation, avoiding catastrophic forgetting (e.g., mitigated in DQN by experience replay), and tuning hyperparameters like learning rates to stabilize training. Function approximation is essential for real-world RL but requires careful design to avoid instability or biased estimates.